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  <div class="section" id="method-and-theory">
<h1>Method and Theory<a class="headerlink" href="#method-and-theory" title="Permalink to this headline">¶</a></h1>
<p>First, an initial system is generated. This is accomplished by</p>
<ol class="arabic simple">
<li><p>Starting with a unit cell system.</p></li>
<li><p>Generating a transformed system by rotating the unit cell such that
the new system’s box vectors correspond to crystallographic
directions, and filled in with atoms to remain a perfect bulk cell
when the three boundaries are periodic.</p></li>
<li><p>All atoms are shifted by a fractional amount of the box vectors if
needed.</p></li>
<li><p>A supercell system is constructed by combining multiple replicas of
the transformed system.</p></li>
</ol>
<p>Two LAMMPS simulations are then performed that apply an energy/force
minimization on the system, and the total energy of the system after
relaxing is measured, <span class="math notranslate nohighlight">\(E_{total}\)</span>. In the first simulation, all of
the box’s directions are kept periodic (ppp), while in the second
simulation two are periodic and one is non-periodic (ppm). This
effectively slices the system along the boundary plane creating two free
surfaces, each with surface area</p>
<div class="math notranslate nohighlight">
\[A = \left| \vec{a_1} \times \vec{a_2} \right|,\]</div>
<p>where <span class="math notranslate nohighlight">\(\vec{a_1}\)</span> and <span class="math notranslate nohighlight">\(\vec{a_2}\)</span> are the two lattice
vectors corresponding to the periodic in-plane directions.</p>
<p>The formation energy of the free surface, <span class="math notranslate nohighlight">\(E_{f}^{surf}\)</span>, is
computed in units of energy over area as</p>
<div class="math notranslate nohighlight">
\[E_{f}^{surf} = \frac{E_{total}^{surf} - E_{total}^{0}} {2 A}.\]</div>
<p>The calculation method allows for the specification of which of the
three box dimensions the cut is made along. If not specified, the
default behavior is to make the <span class="math notranslate nohighlight">\(\vec{c}\)</span> vector direction
non-periodic. This choice is due to the limitations of how LAMMPS
defines triclinic boxes. <span class="math notranslate nohighlight">\(\vec{c}\)</span> vector is the only box vector
that is allowed to have a component in the Cartesian z direction.
Because of this, the other two box vectors are normal to the z-axis and
therefore will be in the cut plane.</p>
</div>


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